Why Does \(\frac{1}{i}\) Equal Both \(i\) and \(-i\)?

[latentcorpse]

i've confused myself

$\frac{1}{i}=-i$ by multiplying by i/i

but then $(\frac{1}{i})^2=-1 \Rightarrow \frac{1}{i}= i,-i$
and clearly the positive root here doesn't make any sense?

 

[Focus]

You are correct but you forgot to take the negative root of the left hand side.

 

[Pengwuino]

Well you've somewhat answered your own question. You simply did pick the incorrect root as there are always 2 roots to a square root

$$\[<br /> \begin{array}{l}<br /> 3 = 3 \\ <br /> 3^2 = 3^2 \\ <br /> 9 = 9 \\ <br /> \sqrt 9 = \sqrt 9 \\ <br /> 3 = - 3? \\ <br /> \end{array}<br /> \]$$

for example.